Properties of Square
A square is a quadrilateral in which all four sides are equal and all four angles are 90 degrees. It is one of the most familiar shapes in geometry. From floor tiles and chessboards to windows and photo frames, the square appears everywhere around us.
A square is a special case of both a rectangle (all angles 90 degrees) and a rhombus (all sides equal). This means a square inherits all the properties of both — making it the most "special" quadrilateral of all.
In Class 8, understanding the properties of a square helps you compare it with other quadrilaterals like rectangles, rhombuses, and parallelograms. You will also use these properties in mensuration to calculate area, perimeter, and diagonal lengths.
What is Properties of Square?
Definition: A square is a rectangle with all four sides equal. Equivalently, a square is a rhombus with all four angles equal to 90 degrees.
A square ABCD satisfies:
- AB = BC = CD = DA (all sides equal)
- angle A = angle B = angle C = angle D = 90 degrees (all angles equal)
- AB || CD and BC || DA (opposite sides parallel)
Hierarchy of Quadrilaterals:
- Every square is a rectangle (all angles 90 degrees)
- Every square is a rhombus (all sides equal)
- Every square is a parallelogram (opposite sides parallel and equal)
- But the reverse is NOT always true
Properties of Square Formula
Formulas for a Square (side = a):
Perimeter = 4a
Area = a²
Diagonal = a√2
Where:
- a = length of each side
- Diagonal formula comes from the Pythagoras theorem: d² = a² + a² = 2a², so d = a√2
Additional formulas:
- Area using diagonal: Area = (1/2) x d² (where d is the diagonal)
- Side from diagonal: a = d / √2
- Side from perimeter: a = Perimeter / 4
- Side from area: a = √(Area)
Derivation and Proof
Deriving the diagonal formula:
Consider square ABCD with side a. Diagonal AC divides the square into two right-angled triangles — triangle ABC and triangle ACD.
In triangle ABC:
- AB = a (one side of the square)
- BC = a (adjacent side of the square)
- Angle B = 90 degrees
Applying Pythagoras Theorem:
- AC² = AB² + BC²
- AC² = a² + a²
- AC² = 2a²
- AC = a√2
Since both diagonals of a square are equal, BD = AC = a√2.
Deriving Area from diagonal:
If diagonal = d, then a = d/√2, so:
- Area = a² = (d/√2)² = d²/2 = (1/2) x d²
Alternatively, since the diagonals of a square are equal and bisect each other at right angles:
- Area = (1/2) x d₁ x d₂ = (1/2) x d x d = d²/2
Types and Properties
The properties of a square can be organised into the following groups:
1. Side Properties:
- All four sides are equal.
- Opposite sides are parallel.
2. Angle Properties:
- All four angles are 90 degrees (right angles).
- Adjacent angles are supplementary: 90 + 90 = 180 degrees.
- Diagonals are equal in length.
- Diagonals bisect each other (cut each other into two equal halves).
- Diagonals bisect each other at right angles (90 degrees).
- Each diagonal bisects the vertex angles (divides the 90-degree angle into two 45-degree angles).
- Each diagonal divides the square into two congruent isosceles right triangles.
- Both diagonals together divide the square into 4 congruent isosceles right triangles.
4. Symmetry Properties:
- A square has 4 lines of symmetry: 2 along the diagonals and 2 through the midpoints of opposite sides.
- A square has rotational symmetry of order 4 (it looks the same after 90, 180, 270, and 360-degree rotations).
5. Relationship with Other Shapes:
- A square is a rectangle with equal sides.
- A square is a rhombus with right angles.
- A square is a parallelogram with equal sides and right angles.
Solved Examples
Example 1: Example 1: Finding perimeter
Problem: Find the perimeter of a square with side 15 cm.
Solution:
Given:
- Side (a) = 15 cm
Using the formula:
- Perimeter = 4a = 4 x 15 = 60 cm
Answer: The perimeter is 60 cm.
Example 2: Example 2: Finding area
Problem: Find the area of a square with side 12 cm.
Solution:
Given:
- Side (a) = 12 cm
Using the formula:
- Area = a² = 12² = 144 cm²
Answer: The area is 144 cm².
Example 3: Example 3: Finding diagonal from side
Problem: Find the diagonal of a square with side 8 cm.
Solution:
Given:
- Side (a) = 8 cm
Using the formula:
- Diagonal = a√2 = 8√2 = 8 x 1.414 = 11.312 cm (approx)
Answer: The diagonal is 8√2 cm (approximately 11.31 cm).
Example 4: Example 4: Finding side from perimeter
Problem: The perimeter of a square is 64 cm. Find the side and the area.
Solution:
Given:
- Perimeter = 64 cm
Finding the side:
- Side = Perimeter / 4 = 64 / 4 = 16 cm
Finding the area:
- Area = side² = 16² = 256 cm²
Answer: The side is 16 cm and the area is 256 cm².
Example 5: Example 5: Finding side from area
Problem: The area of a square is 225 cm². Find the side and the perimeter.
Solution:
Given:
- Area = 225 cm²
Finding the side:
- Side = √Area = √225 = 15 cm
Finding the perimeter:
- Perimeter = 4 x 15 = 60 cm
Answer: The side is 15 cm and the perimeter is 60 cm.
Example 6: Example 6: Finding side from diagonal
Problem: The diagonal of a square is 14√2 cm. Find the side and the area.
Solution:
Given:
- Diagonal = 14√2 cm
Finding the side:
- Diagonal = a√2
- 14√2 = a√2
- a = 14 cm
Finding the area:
- Area = a² = 14² = 196 cm²
Answer: The side is 14 cm and the area is 196 cm².
Example 7: Example 7: Area using diagonal
Problem: The diagonal of a square is 10 cm. Find the area without first finding the side.
Solution:
Given:
- Diagonal (d) = 10 cm
Using the formula:
- Area = (1/2) x d² = (1/2) x 10² = (1/2) x 100 = 50 cm²
Answer: The area is 50 cm².
Example 8: Example 8: Comparing square and rectangle
Problem: A square and a rectangle both have a perimeter of 40 cm. The rectangle has length 12 cm. Which has a greater area?
Solution:
Square:
- Side = 40/4 = 10 cm
- Area = 10² = 100 cm²
Rectangle:
- 2(l + b) = 40, so l + b = 20
- b = 20 - 12 = 8 cm
- Area = 12 x 8 = 96 cm²
Answer: The square has the greater area (100 cm² > 96 cm²). Among all rectangles with a given perimeter, the square has the maximum area.
Example 9: Example 9: Angles formed by diagonals
Problem: In square ABCD, the diagonals AC and BD intersect at O. Find: (a) angle AOB, (b) angle OAB.
Solution:
(a) The diagonals of a square bisect each other at right angles.
- Therefore, angle AOB = 90 degrees.
(b) The diagonal bisects the vertex angle of 90 degrees.
- angle OAB = 90/2 = 45 degrees.
Verification: In triangle AOB: angle OAB + angle OBA + angle AOB = 180. So 45 + 45 + 90 = 180. Verified.
Answer: angle AOB = 90 degrees, angle OAB = 45 degrees.
Example 10: Example 10: Square on the coordinate plane
Problem: A square has vertices at A(0, 0), B(6, 0), and C(6, 6). Find the fourth vertex D and the length of the diagonal.
Solution:
Finding D:
- Since ABCD is a square with AB along the x-axis and BC along the y-axis, D must be at (0, 6).
Finding the diagonal:
- Side = 6 units
- Diagonal = 6√2 units (approximately 8.49 units)
Verification using distance formula for AC:
- AC = √[(6-0)² + (6-0)²] = √(36 + 36) = √72 = 6√2. Verified.
Answer: D = (0, 6), diagonal = 6√2 units.
Real-World Applications
The properties of a square are used in many practical situations:
- Construction: Rooms, floor tiles, window panes, and glass panels are often square-shaped. Knowing perimeter helps calculate material needed for borders, and area helps calculate paint or flooring.
- Land Measurement: Square plots of land are common. Area calculations help determine the cost of land.
- Board Games: Chess and checkerboards are made of small squares. The 8 x 8 grid uses square properties for uniform spacing.
- Pixel Displays: Computer screens and phone displays use square pixels. Each pixel is a tiny square.
- Packaging: Square boxes and cartons are efficient for stacking and storage. Understanding diagonal length helps determine if items fit diagonally inside a box.
- Art and Design: Square canvases, photo frames, and tiles use the symmetry of squares for aesthetic appeal.
- Mathematics: The unit square (side = 1) is the basis for defining area in coordinate geometry.
Key Points to Remember
- A square has all four sides equal and all four angles equal to 90 degrees.
- A square is both a rectangle and a rhombus.
- Perimeter = 4a, Area = a², Diagonal = a√2.
- The diagonals of a square are equal in length.
- The diagonals bisect each other at right angles (90 degrees).
- Each diagonal bisects the vertex angle into two 45-degree angles.
- A square has 4 lines of symmetry.
- A square has rotational symmetry of order 4.
- Among all rectangles with the same perimeter, the square encloses the maximum area.
- Area from diagonal: Area = (1/2) x d².
Practice Problems
- Find the area and perimeter of a square with side 21 cm.
- The perimeter of a square is 84 cm. Find its area.
- The diagonal of a square is 20 cm. Find the side and the area.
- The area of a square is 400 cm². Find the side, perimeter, and diagonal.
- A square has the same perimeter as a rectangle of length 18 cm and breadth 6 cm. Find the side and area of the square.
- Find the area of a square whose diagonal is 7√2 cm.
- How many squares of side 5 cm can fit in a square of side 30 cm?
- A wire 80 cm long is bent into a square. Find the area enclosed.
Frequently Asked Questions
Q1. What is a square?
A square is a quadrilateral with all four sides equal and all four angles equal to 90 degrees.
Q2. Is a square a rectangle?
Yes. A square is a special rectangle where all four sides are also equal. Every square is a rectangle, but not every rectangle is a square.
Q3. Is a square a rhombus?
Yes. A square is a special rhombus where all four angles are 90 degrees. Every square is a rhombus, but not every rhombus is a square.
Q4. How do you find the diagonal of a square?
Diagonal = side x √2. If the side is 10 cm, the diagonal is 10√2 = 14.14 cm (approx).
Q5. How many lines of symmetry does a square have?
A square has 4 lines of symmetry — 2 along the diagonals and 2 through the midpoints of opposite sides.
Q6. What angle does each diagonal make with the sides?
Each diagonal of a square makes an angle of 45 degrees with each side, since the diagonal bisects the 90-degree vertex angle.
Q7. Are the diagonals of a square equal?
Yes. Both diagonals of a square are equal in length. Each diagonal = side x √2.
Q8. Can the area of a square be found using the diagonal?
Yes. Area = (1/2) x diagonal². If the diagonal is 12 cm, the area = (1/2) x 144 = 72 cm².
Q9. What is the difference between a square and a rhombus?
Both have all four sides equal. A square has all angles = 90 degrees and equal diagonals. A rhombus can have angles other than 90 degrees and its diagonals are generally unequal.
Q10. Why does a square have maximum area for a given perimeter?
Among all rectangles with the same perimeter, the square has the maximum area. This is because for a fixed sum (l + b), the product l x b is maximum when l = b.
